The curvatures of the slowness surface for anisotropic media

The curvatures of the slowness surface for anisotropic media

The Gaussian curvature of the slowness surface plays very important role in wave propagation in anisotropic media. It controls the wave amplitudes via the geometrical spreading factor (Gajewski, 1993; Cerveny, 2001; Stovas, 2018; Stovas et al, 2022).

We define the Gaussian and mean curvatures in vicinity of arbitrary point of the slowness surface are convenient to describe in cylindrical coordinate system. If the point on the slowness surface is regular, there is no azimuthal dependence for series coefficients. In case of non-degenerated singularity point (double or triple), all the coefficients in series are azimuthally dependent, and Gaussian and mean curvatures are not defined. For degenerated singularity points, we have only zero-order term which is azimuthally dependent.

We show that one of the principal curvatures can turn to zero at some azimuth angles. In this case, we have the parabolic line (zero Gaussian curvature) associated with singularity point and resulting the caustic in the group velocity domain.

We show examples of parabolic line computed for S1 and S2 waves in vicinity of double (S1&S2) singularity point on the vertical axis with conical and wedge degeneracies (Stovas et al. 2024).

References

Cerveny, V., 2001, Seismic ray theory, Cambridge Univ. Press.

Gajewski, D., 1993, Radiation from point sources in general anisotropic media, Geophysical Journal International, 113(2), 299-317.

Stovas, A., 2018, Geometrical spreading in orthorhombic media, Geophysics, 83(1), C61-C73.

Stovas, A., Roganov, Yu., & V. Roganov, 2022, The S waves geometrical spreading in elliptic orthorhombic media, Geophysical Prospecting 70(7), 1085-1092.

Stovas, A., Roganov, Yu., & V. Roganov, 2024, Singularity points and their degeneracies in anisotropic media, Geophysical Journal International 238 (2), 881-901.